Words, Deeds, and Lies: Strategic Behaviour in Games with Multiple Signals

Transcription

1 Review of Economic Studies (2006) 73, /06/ $02.00 Words, Deeds, and Lies: Strategic Behaviour in Games with Multiple Signals JOHN DUFFY University of Pittsburgh and NICK FELTOVICH University of Houston First version received September 2003; final version accepted October 2004 (Eds.) We report the results of an experiment in which subjects play games against changing opponents. In one treatment, senders send receivers messages indicating intended actions in that round, and receivers observe senders previous-round actions (when matched with another receiver). In another treatment, the receiver additionally observes the sender s previous-round message to the previous opponent, enabling him to determine whether the sender had lied. We find that allowing multiple signals leads to better outcomes when signals are aligned (all pointing to the same action), but worse outcomes when signals are crossed. Also, senders signals tend to be truthful, though the degree of truthfulness depends on the game and treatment, and receivers behaviour combines elements of pay-off maximization and reciprocity. 1. INTRODUCTION Many situations can be modelled as one-shot games in which players interests are at least partially aligned. People in these situations often manage to coordinate successfully on one equilibrium when several exist. They may even obtain higher pay-offs than in any equilibrium (e.g. by overcontributing relative to equilibrium in public good situations). How do they achieve such good outcomes? One promising theory is that they use available information to determine the actions others are likely to choose. 1 This information can take many forms. Players may be able to communicate their intentions with costless non-binding cheap talk. Alternatively, they might use observations of their opponents past behaviour to infer their likely future behaviour. This paper is an examination of these two types of information cheap talk and observation. We design and run an experiment in which subjects play games against changing opponents under two information treatments. In our first, words and deeds (WD) treatment, each sender sends a cheap-talk message to her opponent, indicating the action the sender intends to play. The receiver also observes the sender s previous-round action choice (when matched with someone else). The receiver can use the sender s previous behaviour to make inferences about the likely truthfulness of the sender s message, and from that, the action the sender might choose in the current round. In our second, words, deeds, and lies (WDL) treatment, receivers see the message 1. Another possibility is that some outcome serves as a focal point for players; each player perceives that the other player will choose that outcome, and so they choose it as well. See Sugden (1995) for a first effort in this direction. We consider this theory to be complementary, since the question then becomes one of how players recognize the existence of a focal point; such recognition may be easier if additional information is available for players to use. 669

2 670 REVIEW OF ECONOMIC STUDIES and previous-round action, plus a third piece of information: the sender s previous-round message to her then opponent. Observation of the previous round in this treatment therefore consists of two different pieces of information: what action the sender actually chose and whether the sender had lied about her intentions to the previous receiver. The receiver can use the truthfulness (or lack thereof) of the sender s previous message to evaluate her current message. Our examination consists of two distinct, though related, issues. In both cases, we build upon previous work (Duffy and Feltovich, 2002), which used the same games, but with three different information treatments: cheap talk only, observation of previous-round actions only, and a control treatment with neither cheap talk nor observation. One issue we examine is how the availability of multiple signals affects the outcomes subjects achieve. 2 There are two (not mutually exclusive) possibilities. Giving subjects additional information might enable them to make better decisions, so that good outcomes are more likely. (This is part of the motivation behind our WDL treatment.) On the other hand, allowing multiple signals opens up the possibility that they might be crossed, for example, a current-round message different from the previous-round action. When signals are crossed, the receiver may have no better idea of the sender s likely action than if he had been given no information at all, so the resulting outcome may actually be worse than if only one piece of information had been available. 3 The vast majority of theoretical and experimental treatments of signalling has examined situations with only one signal available. Some theorists have looked at multiple signals, but there has been almost no experimental work designed to see how people actually behave in these situations. (Exceptions are discussed in Section 3.) The other issue we examine deals with the question: Do actions speak louder than words? Our earlier paper addressed this question indirectly, by comparing subject behaviour in the cheaptalk treatment with that in the observation treatment. We found that the answer varied with the game, specifically on whether it gave senders incentives to be truthful with their cheap talk. When the game s structure implied that cheap talk should be credible, cheap talk was more effective than observation words spoke louder than actions. When it implied that cheap talk should not be credible, observation was more effective actions spoke louder than words. However, this earlier paper had the limitation that no direct comparisons between the two signalling devices were possible. Our current paper remedies this deficiency, by giving receivers both pieces of information. Our new treatments allow us to see how receivers weigh each piece. We can also see whether their behaviour is warranted, by examining how senders actual choices are related to the signals they sent in particular, whether actions or words are better indicators of senders current-round behaviour. Additionally, the WDL treatment allows us to ascertain whether behaviour depends on receivers ability to verify senders past truthfulness. Our main findings are as follows. The new, multiple-signal treatments lead to more cooperation, more coordination, and higher pay-offs than when no signals are available (the control treatment). However, they do not improve upon the levels observed when exactly one signal is available (the cheap-talk and observation treatments); in fact, they frequently make matters worse! This aggregate-level result masks sharp differences in outcomes following aligned signal combinations (signals all pointing to the same action) and those following crossed-signal combinations. Outcomes following aligned signals are generally as good as, or better than, those in the one-signal treatments, while outcomes following crossed signals are worse, and comparable to those in the no-signal treatment. Even after accounting for changes in levels of cooperation, successful coordination is more likely, and average pay-offs higher, after aligned signals than 2. For ease of exposition, we use the term signal to encompass both messages and observed actions, and more generally, any piece of information sent to another player, intentionally or not, as long as both sender and receiver know that it is sent and received. We note here that all signals in our experiment are costless and non-binding. 3. As an old expression goes, the man with one watch always knows what time it is, while the man with two watches never does.

3 DUFFY & FELTOVICH WORDS, DEEDS, AND LIES 671 FIGURE 1 The games TABLE 1 Characteristics of Nash equilibria of the games Game Equilibrium P(Cooperate) P(Coordinate) Expected pay-offs Pay-off efficiency Efficiency gain PD (0, 0) 0 (40,40) (1, 1) 1 1 (70,70) SH (0, 0) 0 1 (55,55) (0 75, 0 75) (55,55) (1, 0) (50,80) CH (0, 1) (80,50) (0 5, 0 5) (60,60) Nash equilibria are presented in the form (P(row player cooperates), P(column player cooperates)). PD, Prisoners Dilemma; SH, Stag Hunt; CH, Chicken. after crossed signals. In other words, the amount of information received is generally less important than the content of that information. At a more disaggregated level, we find that senders actions are positively correlated with their current-round messages and previous-round actions. In particular, messages tend to be truthful, even when the structure of the game provides incentives to lie, though truthfulness varies with the game and increases in the WDL treatment, where lies are detectable. Receivers, for their part, use the information they receive in a way that combines pay-off maximization with reciprocity (receivers cooperate more with senders who they deem likely to cooperate). 2. THE GAMES We use three games: Prisoners Dilemma (PD), Stag Hunt (SH), and Chicken (CH) (see Figure 1). Each game has two strategies, which we label Cooperate (C) and Defect (D). These games were chosen because they are well-known, symmetric, 2 2 games in which choosing C always weakly increases the other player s pay-off. Desirable outcomes for these games (from the players standpoint) include cooperation, coordination on a pure-strategy Nash equilibrium, and high pay-offs. By good outcomes we refer to outcomes in which as many as possible of these features are present. Table 1 reports the extent to which they are present in the Nash equilibria of these games. The overall frequency of C choices is found under the heading P(Cooperate). The probability of coordination, P(Coordinate), refers to the likelihood that players play a pure-strategy equilibrium when the game has multiple equilibria: (C,C) or (D,D) in Stag Hunt, and (C,D) or (D,C) in Chicken. Two measures of efficiency are used. Pay-off efficiency is the sum of the pay-offs of the row and column players, normalized so that the maximum possible joint pay-off in a given game has an efficiency of 1 and the minimum has an efficiency of 0. Efficiency gain is similar, except that it is the lowest-pay-off Nash equilibrium whose efficiency is defined as 0 (so that negative values are possible).

4 672 REVIEW OF ECONOMIC STUDIES 3. THE INFORMATION TREATMENTS: THEORY AND HYPOTHESES From a theoretical standpoint, allowing signals need not affect outcomes in the games we consider. Because the games are finitely repeated, and this fact is public knowledge, the set of subgame-perfect equilibria corresponds closely to the set of sequences of stage-game Nash equilibria. Allowing current-round messages does not increase the number of action sequences consistent with equilibrium, though the number of equilibria may increase due to the possibility of players conditioning their actions on the message. Previous-round actions (by themselves, or in conjunction with previous-round messages), on the other hand, do open the possibility of equilibrium play containing action profiles that are not stage-game Nash equilibria: for example, (C,C) in Chicken. 4 Aumann (1990) and Farrell and Rabin (1996) propose conditions for messages to be truthful in situations where messages have literal meanings (i.e. some convention exists for translating each message into a unique intended action), as they do in our design. We adopt Farrell and Rabin s nomenclature here. Their conditions make use of the reasonable assumption that, if the receiver believes the sender s message to be truthful (the same as the sender s subsequent action), he will choose an action that is a best response to that message. The first condition, selfcommitment, is satisfied when the sender s message is, in turn, a best response to the receiver s action (so they form a Nash equilibrium). The second condition, self-signalling, is satisfied when (a) a sender intending to be truthful prefers the receiver to play his best response, and (b) a sender who intended her message to be deceptive would not prefer the receiver to play his best response. In Stag Hunt, both C and D messages are self-committing and self-signalling; in Chicken, both are self-committing but not self-signalling; and in Prisoners Dilemma, C messages are neither self-committing nor self-signalling, while D messages are self-committing but not self-signalling. Following Farrell and Rabin, who note that a message that is both self-signalling and selfcommitting seems highly credible (p. 112), we therefore expect that messages in Stag Hunt should most often be truthful and believed, messages in Chicken and D messages in Prisoners Dilemma should less often be truthful and believed, and C messages in Prisoners Dilemma should least often be truthful and believed. By contrast with cheap-talk messages, observed previous-round actions are credible by their very nature. However, they also differ from messages in the extent to which they can be considered signals of the sender s likely current-round action. There is no question that a message is a signal; that is its only function. In contrast, the observer of a previous-round action must bear in mind that it plays a dual role of signal for the current round and action choice for the previous round. This is equally true in each of our three games, so we expect that the correlation between previous-round actions and current-round actions will not vary systematically with the game. Therefore, we hypothesize that the efficacy of observation vs. cheap talk in facilitating good outcomes should depend simply on how credible cheap talk is in our games. When cheap talk is relatively credible, it should be more effective than observation; when cheap talk is relatively incredible, it should be less effective than observation. In addition to our earlier paper (Duffy and Feltovich, 2002), which gave evidence broadly supporting this hypothesis, we know of two previous papers comparing cheap talk and 4. One way this can happen is as described by Okuno-Fujiwara and Postlewaite (1995), who look at infinitely repeated games with discounting and random matching. In their model, each player carries a status with her, which is observable to opponents and can be updated in response to her actions. Her status functions as a proxy for her history of play, so that punishment strategies are possible even when players play each other only once. The resulting norm equilibria correspond to subgame-perfect equilibria in standard infinitely repeated games (with fixed opponents); in particular, mutual cooperation can be enforced. In our set-up, players play only finitely many times, so mutual cooperation is not possible in an equilibrium in Prisoners Dilemma; however, it is possible in Chicken.

5 DUFFY & FELTOVICH WORDS, DEEDS, AND LIES 673 observation. 5 Wilson and Sell (1997) examined cheap talk and observation in a public-good game, in which the same group of subjects repeatedly chose contributions. They found that combining cheap talk and observation of past contributions resulted in contribution levels approximately the same as when neither cheap talk nor observation was present. However, either cheap talk alone or observation alone actually led to decreased contributions, so that there were substantial social returns to adding the second type of signal, given that the first was already present. Their experimental set-up was substantially different from ours, however (e.g. their control treatment gave subjects no feedback at all, while our subjects learned their opponents current-round actions after they took place), so their results do not carry much implication for our experiment. Çelen, Kariv and Schotter (2005) examined observation and cheap talk in an information cascade experiment, where incentives were such that message senders interests were closely aligned with receivers. (Senders received a payment if the receiver guessed correctly.) They found that messages tended to be truthful and believed, that allowing cheap talk improved pay-offs much more than allowing observation, and that allowing observation on top of cheap talk made little further improvement (though adding cheap talk on top of observation did improve pay-offs). Because this was a situation in which messages were expected to be extremely credible, their results are encouraging. In this paper, we go even further, examining the role of signals in strategic environments where cheap-talk signals need not be so credible. 4. EXPERIMENTAL PROCEDURES We used a 3 2 design in which we varied the order in which the games were played (PD SH CH, SH CH PD, or CH SH PD) and the information condition (WD or WDL). 6 Each experimental session involved 20 subjects playing 10 rounds of each game under a single information condition. Subjects were primarily University of Pittsburgh undergraduates. No subject participated in more than one session, nor did any who participated in the experiment of our previous paper participate in this experiment. In each game, 10 of the subjects were row players, and the other 10 were column players. Subjects were randomly assigned one of these roles and remained in the same role throughout a game. Each row player faced each column player exactly once in each game. 7 Sessions were conducted in the Pittsburgh Experimental Economics Laboratory, using networked personal computers. Each subject was seated at a computer and was given written instructions. These instructions were also read aloud in an effort to make the rules of the experiment common knowledge. The computer screen displayed the current game s pay-off matrix, the results of the player s own previous rounds of play of that game, and signals sent or received. 5. There are many papers that examine either cheap talk or observation alone. Papers that examine cheap talk include Cooper, Dejong, Forsythe and Ross (1989, 1992), Charness (2000), Blume and Ortmann (2006), and Burton, Loomes and Sefton (2006). See also Crawford (1998) for a survey of experiments involving cheap talk. Papers that examine observation include Kahneman, Knetsch and Thaler (1986), Eckel and Grossman (1996), Fehr, Gächter and Kirchsteiger (1997), Duffy and Feltovich (1999), Huck, Normann and Oechssler (1999, 2000), Bosch-Doménech and Vriend (2003), and Simonsohn, Karlsson, Loewenstein and Ariely (2004). 6. We actually split our WD treatment into random (WDr) and nonrandom (WDnr) subtreatments. In WDr, the roles of sender and receiver were determined randomly at the beginning of each round, as in our previous experiment. In WDnr, roles were determined randomly prior to the first round of a game and remained the same for all 10 rounds played. This was done so that we could make direct comparisons with data from other treatments. The WDr treatment was chosen to facilitate comparison with the treatments from the previous experiment, while the WDnr treatment was chosen to facilitate comparison with the WDL treatment, where the presence of previous-round messages made it necessary to fix the roles of sender and receiver in all rounds of a game. As it turned out, we were unable to find any differences between the WDr and WDnr data; therefore, we simply pooled these data. 7. Kamecke (1997) shows that the matching technique we used, rotation, ensures that the 10-round game maintains the one-shot character of the stage game, and does so efficiently in the sense that there is no way to increase the number of rounds, while keeping the same number of players and continuing to maintain the one-shot nature of the game.

6 674 REVIEW OF ECONOMIC STUDIES The current pay-off matrix was also drawn on a blackboard for all to see. Subjects input their actions by choosing the row or column of the pay-off matrix they wanted to play. Row players actions were labelled R1 and R2, and column players were labelled C1 and C2, in both cases corresponding to C and D, respectively. In describing the actions to subjects we avoided reference to the labels cooperate or defect, and we referred to a player s opponent as his or her partner. Also, subjects were not given advice about how they should make use of the information they were given; for example, subjects in the WDL treatment were not told that previous-round actions might be used to assess the truthfulness of previous-round messages. No signals were received in round 1 of each game. 8 In rounds 2 10, cheap talk and observation of previous-round actions took place before subjects chose their current-round actions. Observation of previous-round messages in the WDL sessions also took place at this time, from the third round on. After all subjects had chosen their actions for the current round, each was informed of her pay-off and her opponent s action in that round. Each point in the pay-off matrix represented a 1% chance of winning $1 00. At the end of every round, an integer between 1 and 100 inclusive was randomly drawn. Subjects whose pay-off in that round was greater than or equal to the chosen number earned $1 00 for the round; those whose pay-off was lower earned nothing for the round. 9 At the end of the session, subjects received in cash their total earnings from all rounds, as well as a $5 show-up fee. Subjects earned an average of about $25; sessions typically lasted between 60 and 75 minutes. 5. RESULTS The experiment consisted of six sessions each of the WD and WDL treatments; within either treatment, there were two sessions using each of the three orderings. 10 We address the issue of differences in play due to changes in the ordering of the games in Sections 5.3 and 5.4; for now, we pool the data from sessions with different orderings Population aggregates Table 2 shows treatment-wide levels of cooperation, coordination, and both measures of efficiency. For comparison, we include corresponding results from the cheap-talk-only, observationonly, and control (neither cheap talk nor observation) treatments of Duffy and Feltovich (2002). Superscripts in the table refer to significance of differences between two information treatments; for a given game and statistic, entries sharing a superscript are not significantly different, while entries with letters earlier in the alphabet correspond to significantly lower values. 11 (E.g. a statistic with a b superscript is significantly higher than one with an a superscript, but neither is significantly different from one with an ab superscript.) In our previous paper, we found that allowing either cheap talk or observation resulted in higher levels of cooperation, coordination, and pay-off efficiency than we saw in the control. 8. In the first round, observation of past actions is not possible. To maintain the symmetry of treatment of the two types of signal, we chose to suspend cheap talk in the first round as well. An implication of this design feature is that, in the WDL cell, previous-round messages could not be observed until the third round. 9. This binary lottery procedure is intended to induce risk-neutral behaviour among hypothetical expected utility maximizing agents. See, for example, Roth and Malouf (1979) for a discussion. 10. The instructions used in the experiment and the raw data are available from the authors upon request. 11. We use the robust rank-order test instead of the more commonly used Wilcoxon Mann Whitney test because the latter assumes that the two samples being compared come from distributions with identical second- and higherorder moments, which we have no reason to believe a priori. See Siegel and Castellan (1988) for a discussion of this issue, as well as more thorough descriptions of the non-parametric statistical tests used in this paper. See Feltovich (2003) for a simulation-based comparison of the robust rank-order and Wilcoxon Mann Whitney tests, under a variety of distributional assumptions. All of our non-parametric statistical tests are performed on data at the session level.

7 DUFFY & FELTOVICH WORDS, DEEDS, AND LIES 675 TABLE 2 Relative frequencies of cooperation, coordination, and efficiency (all rounds) Game Treatment Cooperation Coordination Pay-off efficiency Efficiency gain WD ab ab ab WDL b b b PD Cheap talk b b b Observation b b b Control a a a WD ab bc abc abc WDL b c c c SH Cheap talk ab c bc bc Observation ab ab ab ab Control a a a a WD ab ab ab ab WDL ab ab ab ab CH Cheap talk ab b ab ab Observation b a b b Control a a a a Italicized treatments from Duffy and Feltovich (2002). Within each game and statistic, entries with no superscripts in common are significantly different at the 10% level (two-sided robust rank-order test, session-level data); superscripts earlier in the alphabet correspond to significantly lower values. PD, Prisoners Dilemma; SH, Stag Hunt; CH, Chicken; WD, Words and Deeds; WDL, Words, Deeds, and Lies. Table 2 shows that allowing both cheap talk and observation (the WD treatment) almost never significantly improves outcomes over the control, observation-only, or cheap-talk-only treatments, and outcomes are often significantly worse than in one or the other of these one-signal treatments. Adding observation of previous-round messages (the WDL treatment) improves matters a bit: outcomes are never significantly worse than in any other treatment and sometimes significantly better than in the control and observation treatment. However, they are never significantly better than those in either the cheap-talk treatment or the WD treatment. Thus, while the value of allowing signalling is high if none is currently allowed, incremental social returns to additional signals are small or even negative. One explanation for this finding is that it is not simply the amount of information that matters, but rather its content. 12 When players have access to only one signal, its interpretation is unambiguous (though not necessarily truthful). With more than one signal, however, the potential exists for signal combinations with no clear implication, such as a C message and D observed action. Hence, we must distinguish between aligned (all C or all D ) and crossed (at least one C and one D ) signal combinations. Aligned-signal combinations can be interpreted as a single signal, or even as a signal that s been reinforced. On the other hand, crossed-signal combinations may contain little or no information value. 13 We address this issue in Figure 2, which shows the levels of cooperation, coordination, and pay-off efficiency for rounds 2 10 of the WD and WDL treatments of each game, broken 12. Another possible explanation is the phenomenon of information overload (Earl, 1990): as the amount of information provided increases, decision-makers tend to devote less careful attention to deciding whether they need to make changes to their strategies. 13. In the WDL treatment, some crossed-signal combinations may carry information. For example, a receiver viewing the combination of D previous-round message, D previous-round action, and C current-round message may reason that, because the sender was truthful in the previous round, she will be truthful in the current round also, and therefore her current-round action should be C. In Section 5.4, we look at behaviour following such signal combinations in the WDL treatment. For now, we simply point out that, even if not completely uninformative, they are likely to be less informative than aligned-signal combinations.

8 676 REVIEW OF ECONOMIC STUDIES PD, Prisoners Dilemma; SH, Stag Hunt; CH, Chicken. FIGURE 2 Relative frequencies of cooperation, coordination and efficiency (rounds 2 10) down according to whether the signals received were aligned or crossed. Also shown are the overall levels of cooperation, coordination, and efficiency for the WD and WDL treatments, as well as the control, cheap-talk, and observation treatments from Duffy and Feltovich (2002). In the PD and SH games, levels of cooperation, coordination, and efficiency are always significantly higher when signals are aligned than when signals are crossed. (Indeed, if WD and WDL data are pooled, the difference is always significant at the 1% level.) Moreover, when signals are aligned, outcomes are comparable to those from the earlier observation and cheap-talk treatments whose single signals are, by definition, aligned but when signals are crossed, levels of cooperation, coordination, and efficiency are much lower and comparable to those from the control treatment. The CH results are more complicated. There are no significant differences (p > 0 10 for WD alone, WDL alone, and both together) in either cooperation or efficiency between aligned- and crossed-signal combinations. In fact, they are sometimes actually slightly (though not significantly) lower after aligned signals than after crossed signals. Coordination in the CH WD cell after aligned-signal combinations is significantly higher than after crossed-signal combinations, and is comparable to coordination in the earlier observation and cheap-talk treatments. However, there is no such significant difference in the CH WDL cell. (If CH WD and CH WDL data are pooled, coordination is significantly higher at the 5% level after aligned signals than after crossed signals.) Overall, aligned-signal combinations lead to especially high levels of cooperation, coordination, and efficiency, while crossed-signal combinations lead to little or no improvement over no signals at all. This suggests that the overall lack of improvement from the earlier cheap-talk and observation treatments to the WD and WDL treatments, as seen in Table 2, may mask better outcomes following aligned-signal combinations and worse outcomes following crossed-signal combinations A closer look at coordination and efficiency We saw in Table 2 and Figure 2 that coordination and efficiency tended to be high following aligned-signal combinations, but not following crossed-signal combinations. We will see now that these levels are high not only in an absolute sense, but also relative to what would have been expected given the observed frequency of cooperation. To see why this might be so, we first examine correlation in players actions. If players are unable to make use of the extra information available to them, then sender and receiver actions should be uncorrelated. Table 3 shows, on the contrary, that there typically is some correlation; it

9 DUFFY & FELTOVICH WORDS, DEEDS, AND LIES 677 TABLE 3 Correlation coefficients of sender and receiver actions (rounds 2 10) Correlation Game Treatment Overall After aligned signals After crossed signals PD WD WDL SH WD WDL CH WD WDL PD, Prisoners Dilemma; SH, Stag Hunt; CH, Chicken; WD, Words and Deeds; WDL, Words, Deeds, and Lies. is positive in Prisoners Dilemma and Stag Hunt, and negative in Chicken. This table also shows correlation coefficients following aligned- and crossed-signal combinations. In both PD and both SH cells, correlation of sender and receiver actions is more positive after aligned-signal combinations than after crossed-signal combinations, while in both CH cells, this correlation is more negative after aligned-signal combinations than after crossed-signal combinations. (Indeed, in the SH WD and CH WD cells, the correlation following crossed signals actually has the opposite sign.) The correlation between sender and receiver actions, and the difference in correlation between that following aligned signals and that following crossed signals, have effects on coordination and efficiency. Specifically, coordination and efficiency are high in these games even accounting for the overall observed levels of cooperation and the increase is large following aligned signals and small (or non-existent) following crossed signals. First, consider coordination. In Stag Hunt, coordination means play by the two players of either a (C,C) or a (D,D) strategy pair. Hence, if q 1 and q 2 denote the observed frequencies of cooperation in a SH session (by senders and receivers, respectively), then we can define the predicted frequency of coordination to be q 1 q 2 + (1 q 1 )(1 q 2 ). 14 In Chicken, coordination means play of either a (C,D) or a (D,C) pair, so that if r 1 and r 2 are the observed frequencies of cooperation in a CH cell, the predicted frequency of coordination will be r 1 (1 r 2 ) +r 2 (1 r 1 ). Next, consider efficiency. Using our pay-off efficiency measure, the efficiency of a (C,C) pair in Prisoners Dilemma is 1, that of (D,D) is 0, and that of (C,D) or (D,C) is 1 6. So, if s 1 and s 2 are the observed frequencies of cooperation, we define the predicted levelofefficiency to be s 1 s [s 1(1 s 2 ) + (1 s 1 )s 2 ]. Similarly, predicted levels of efficiency are q 1 q (1 q 1)(1 q 2 ) in Stag Hunt and r 1 r [r 1(1 r 2 ) + (1 r 1 )r 2 ] in Chicken. Figure 3 plots the predicted and observed levels of coordination for Stag Hunt and Chicken and efficiency for all three games, broken down by session and by whether signals were aligned or crossed. Also shown is the 45 line, where predicted and observed levels are equal. Following aligned signals, observed coordination and efficiency are typically higher than would be predicted based on the observed levels of cooperation; this is true both overall (one-sided Wilcoxon summed-ranks test, p < 0 001) and for each game individually (p < 0 01). 15 On the other hand, 14. There is some room for confusion here, since predicted can also be used to refer to the theoretical (Nash equilibrium) levels of coordination and efficiency, shown in Table 1. We emphasize that the predicted levels of coordination and efficiency discussed in this section are derived from the observed levels of cooperation by sender and receiver, the assumption of zero correlation between sender and receiver actions, and (in the case of efficiency) the game s pay-off matrix. 15. If anything, Figure 3 understates the differences between observed and predicted valued following aligned signals, due to the fact that, for a given game and given levels of sender and receiver cooperation, upper bounds on

10 678 REVIEW OF ECONOMIC STUDIES PD, Prisoners Dilemma; SH, Stag Hunt; CH, Chicken. FIGURE 3 Predicted and observed coordination and efficiency, session-level data, rounds 2 10 there are no significant differences between predicted and observed efficiency following crossed signals either overall or for either Stag Hunt or Chicken individually ( p > 0 10), though observed efficiency in Prisoners Dilemma following crossed signals is significantly higher than predicted (p < 0 05) Individual behaviour I: do actions speak louder than words? The results presented above imply that senders and receivers condition their actions on the signals sent and received. We now examine how they do this. Table 4 shows senders and receivers relative frequencies of cooperation, conditional on the current-round message and observed action. Several patterns stand out. First, senders signals both messages and observed actions are quite useful in forecasting their current-round actions. In fact, both types of signal tend to be truthful. 16 For all games and both information treatments and for both messages and observed actions, senders are substantially, and significantly, more likely to choose C after sending a C signal than after sending a D signal. It is perhaps not surprising that senders previous-round actions are truthful, as this may simply mean that players action choices are positively autocorrelated (as indeed they are, both senders and receivers ). What is surprising is that cheap-talk messages are so truthful, even in the PD cells where messages were not expected to be credible. Table 5 shows the overall frequencies of truthful messages, observed actions, and aligned pairs (currentround message and observed action that are either both C or both D) for each cell. The overall frequency of truthful messages varies substantially across games and information treatments, but is well over half in all cells, even in the PD cells. Notice also that aligned pairs are even more truthful than either messages or observed actions alone. If receivers understand how senders signals correlate with their subsequent actions, they ought to condition their own actions on these signals. Indeed, they do so. However, while senders behaviour can be concisely described as truthfulness, receivers respond to senders signals in a complex manner that combines aspects of reciprocity and best response. By reciprocity, coordination and efficiency are typically below 1. For example, in Stag Hunt, given q 1 and q 2 (sender and receiver levels of cooperation), the maximum possible frequency of coordination is (1 + q 1 + q 2 2)max{q 1,q 2 }. This is less than 1 when q 1 and q 2 are different, but close to 1 when they are close to each other, as they typically are in the experimental results. However, in some cases, these upper bounds can be substantially below 1; for example, upper bounds for efficiency after aligned signals in Prisoners Dilemma sessions vary from about 21% to about 52%. 16. We call a signal truthful if it matches the subsequent current-round action; for example, a C current-round message is truthful if the current-round action is also C. It seems uncontroversial to use this term for current-round messages, but there is a slight abuse of vocabulary in using it to refer to an observed previous-round action that matches the current-round action, since as mentioned in Section 3, observed actions have other purposes besides signalling.

11 DUFFY & FELTOVICH WORDS, DEEDS, AND LIES 679 TABLE 4 Relative frequencies of cooperation, conditional on signals sent or received (rounds 2 10) PD WD PD WDL SH WD SH WDL CH WD CH WDL Senders C message (176/392) (212/384) (402/437) (427/464) (218/291) (234/289) D message (20/148) (24/156) (23/103) (19/76) (75/249) (47/251) C obs. action (125/202) (197/255) (391/417) (429/317) (219/317) (237/293) D obs. action (71/338) (39/285) (34/123) (17/95) (74/223) (44/247) Receivers C message (149/392) (143/384) (399/437) (427/464) (167/291) (189/289) D message (17/148) (25/156) (24/103) (21/76) (158/249) (170/251) C obs. action (98/202) (108/255) (377/417) (414/445) (181/317) (193/293) D obs. action (68/338) (60/285) (46/123) (34/95) (144/223) (166/247) C is significantly more likely after a C than after a D signal (one-tailed Wilcoxon summed-ranks test, session-level data, p < 0 10). C is significantly more likely after a C than after a D signal (one-tailed Wilcoxon summed-ranks test, session-level data, p < 0 05). obs., observed; PD, Prisoners Dilemma; SH, Stag Hunt; CH, Chicken; WD, Words and Deeds; WDL, Words, Deeds, and Lies.

12 680 REVIEW OF ECONOMIC STUDIES TABLE 5 Senders frequencies of truthful words, deeds, and aligned pairs WD treatment WDL treatment Game Words Deeds Aligned pairs Words Deeds Aligned pairs PD SH CH PD, Prisoners Dilemma; SH, Stag Hunt; CH, Chicken; WD, Words and Deeds; WDL, Words, Deeds, and Lies. we mean choosing a cooperative action (C in our games) when matched with a sender who is deemed likely to choose a cooperative action and choosing an uncooperative action (D) when the sender is expected to choose an uncooperative action. In our games, reciprocity therefore implies that the receiver chooses the same action as the one he expects the sender to choose. Best responses, on the other hand, vary by the game (recall Figure 1). In Prisoners Dilemma, the best response to either action is D; in Stag Hunt, the best response to either action is the same action; in Chicken, it is the opposite action. Because senders current-round messages and previous-round actions tend to be truthful, we can combine the predictions of reciprocity and pay-off maximization to predict how receivers should react to senders signals. In Prisoners Dilemma, pay-off maximization implies no difference between responses to C signals and those to D signals, so that the only effect is that of reciprocity. Receivers should tend to choose C more often in response to C signals than D signals. In Stag Hunt, reciprocity and pay-off maximization point in the same direction, so their effects reinforce each other. Receivers should tend to choose C in response to C signals more often than in response to D signals, possibly even more so than in Prisoners Dilemma. In Chicken, pay-off maximization and reciprocity point in opposite directions, so will work to cancel each other out. We should thus expect little or no difference between responses to C signals and to D signals. This is exactly what Table 4 shows. In the SH and PD cells, receivers are substantially (and significantly) more likely to cooperate after seeing a C signal than after seeing a D signal. In the CH cells, on the other hand, there is no strong relationship between signal and receiver action; receivers actually tend to choose the opposite action slightly more than the same action. In order to assess the effects of combinations of signals, we next estimate probit regressions in which the binary dependent variable is whether a player chose to play C. We focus for now on the role played by two-signal combinations of current-round message and previousround action, in both the WD and WDL treatments. (Later, in Section 5.4, we add three-signal combinations to our regressions using data from the WDL treatment only.) Independent variables include a constant, the round number, a dummy variable wdl equal to 1 if the data came from the WDL treatment, dummies shfirst and chfirst equal to 1 if the order of the games played was SH CH PD or CH SH PD, and paydiff, a measure of the difference in expected pay-off between a C choice and a D choice. 17 Finally, we included dummies CC, CD, and DD that were equal to 1 whenever the (message, observed action) combination was CC, CD, or DD. 18 We 17. Expected pay-off differences were calculated separately for each game, session, and round as follows. First, we found the relative frequencies of C and D choices in that game and session over all previous rounds. Treating this pair of relative frequencies as a population mixed strategy, we then calculated the expected pay-offs to C and D vs. an opponent using that mixed strategy. The expected pay-off difference was the expected pay-off to C minus the expected pay-off to D. 18. To avoid perfect collinearity, we left out the DC dummy. Analogously, we do not have one for the PD SH CH game ordering.

13 DUFFY & FELTOVICH WORDS, DEEDS, AND LIES 681 also estimated this model under the restriction that these signal-combination dummies were all 0. Both restricted and unrestricted models had subject random effects, to account for unobserved heterogeneity across subjects. 19 Coefficient estimates are shown in Table 6, along with S.E. in parentheses. We also show log-likelihoods (for each regression) and p-values from tests of joint significance of the signalcombination dummies (for each pair of regressions). The results show several regularities. The coefficients for the signal combinations are jointly significant in each case, and are often significant when considered individually. (The main exception is in the regression for Chicken receivers, where no signal coefficient is individually significant, though the three together are jointly significant at the 10% level.) The signs on the signal coefficients reveal that the CC combination tends to have a positive effect on cooperation, while the DD combination has a negative effect. Interestingly, the CD coefficient is significant and positive for senders in all three games; that is, cooperation by senders is higher following a C message and D observed action than following a D message and C observed action (the baseline case) even in Prisoners Dilemma, where cheap talk should not be credible. For receivers, on the other hand, the coefficient for the CD dummy is significant and positive in Stag Hunt, but not in the other two games. So, while senders words speak louder than their actions, receivers are only listening in one game of the three. Since the signal-combination coefficients are jointly significant in every case, we use the unrestricted regressions for our discussion of the other variables. (Results are usually qualitatively similar in the restricted regressions; see Table 6.) The effect of the round number varies. In Prisoners Dilemma, it is significant and negative, consistent with many other studies showing declining cooperation over time. In Stag Hunt, it is insignificant for senders, but significant and positive for receivers. In Chicken, it is insignificant for both senders and receivers. The coefficient on the WDL dummy is never significant, suggesting that behaviour in the WD and WDL treatments is similar, once other factors (such as the signal combinations) are controlled for. Coefficients for the game-order dummies are seldom significant. Coefficients for pay-off difference are significant in four cases out of six; when significant, they are positive except for receivers in Chicken. A positive coefficient is consistent with pay-off maximization, to the extent that our pay-off difference variable reflects true expected pay-offs. A negative coefficient in Chicken is not completely surprising; as mentioned earlier, this is the one game in which reciprocity and pay-off maximization point in opposite directions, so the negative sign suggests simply that reciprocity is relatively powerful here Individual behaviour II: lies, damned lies, and statistics We saw in Section 5.1 that aggregate behaviour in the WDL treatment was somewhat different from that in the WD treatment: cooperation and coordination were more likely and average pay-offs higher (though differences were usually not significant). There are two primary ways in which the addition of information about previous-round messages might lead to improvements in outcomes. It could be that receivers in the WDL treatment who, unlike receivers in the WD treatment, can judge the veracity of senders previous-round messages, are better able to evaluate senders current-round messages and choose their own actions accordingly. A second possibility is that senders, knowing their truthfulness will be observed in the next round, choose their 19. As a robustness check, we also looked at versions of these models with session random effects, with no random effects but with S.E. corrected for clustering within subjects, and with no random effects and S.E. corrected for clustering within sessions. None of these alternative specifications substantially affected our main results.

14 682 REVIEW OF ECONOMIC STUDIES TABLE 6 Random-effects probit results frequency of cooperation (rounds 2 10) Senders (N = 1080) Receivers (N = 1080) Prisoners Dilemma Stag Hunt Chicken Prisoners Dilemma Stag Hunt Chicken WD WDL WD WDL WD WDL WD WDL WD WDL WD WDL const (0 964) (1 042) (0 315) (0 304) (0 261) (0 328) (1 040) (0 919) (0 328) (0 165) (0 225) (0 197) round (0 026) (0 028) (0 031) (0 032) (0 021) (0 024) (0 027) (0 024) (0 028) (0 019) (0 018) (0 018) wdl (0 181) (0 280) (0 215) (0 251) (0 196) (0 382) (0 218) (0 190) (0 230) (0 119) (0 165) (0 165) shfirst (0 229) (0 423) (0 365) (0 409) (0 259) (1 952) (0 282) (0 241) (0 371) (0 195) (0 215) (0 215) chfirst (0 212) (0 299) (0 235) (0 317) (0 226) (0 437) (0 256) (0 221) (0 267) (0 137) (0 193) (0 193) paydiff (0 048) (0 056) (0 016) (0 019) (0 044) (0 120) (0 051) (0 046) (0 018) (0 038) (0 0381) (0 038) CC (0 224) (0 278) (0 170) (0 227) (0 268) (0 137) CD (0 235) (0 324) (0 217) (0 223) (0 273) (0 173) DD (0 242) (0 326) (0 210) (0 248) (0 292) (0 142) ln(l) p-value a < < < < < Note: Signal combinations are (current-round message, previous-round action). *Coefficient is significantly different from 0 at the 10% level. **Coefficient is significantly different from 0 at the 5% level. ***Coefficient is significantly different from 0 at the 1% level. a Likelihood-ratio test of no difference between unrestricted and restricted models. WD, Words and Deeds; WDL, Words, Deeds, and Lies.

15 DUFFY & FELTOVICH WORDS, DEEDS, AND LIES 683 messages and actions differently. (These possibilities are not mutually exclusive.) In this section, we look at these possibilities. Table 7, which shows frequencies of sender and receiver C choices conditional on each three-signal combination, gives some evidence that information about the sender s past truthfulness is useful in predicting her current actions, above and beyond the information present in the current-round message and previous-round action, and that this information seems to be acted on by receivers. Both senders and receivers are most likely to cooperate after a CCC combination and least likely to cooperate after a DDD combination; the lone exception is for receivers in Chicken, where best response and reciprocity pull in opposite directions. A few other patterns can be seen in this table, if we combine similar types of signal combination. We classify the eight possible three-signal combinations into three classes: (1) truth, where the previous-round message and previous-round action were the same (combinations 1, 3, 6, and 8); (2) nice lie,ad previous-round message and a C previous-round action (combinations 5 and 7); and (3) damned lie, a C previous-round message and a D previous-round action (combinations 2 and 4). The strongest pattern is that following truth, the sender is likely to be truthful again: her currentround action is likely to be the same as her current-round message. This happens with frequency 76 6% in PD sessions, 97 1% in SH sessions, and 90 8% in CH sessions. 20 Less strong, but still discernable, are the patterns following lies. Following a nice lie, senders tend to choose C; this happens with frequency 57 1% in PD sessions, 70 0% in SH sessions, and 78 0% in CH sessions. Following a damned lie, senders tend to choose D; they choose C only 10 1% of the time in PD sessions, 7 7% of the time in SH sessions, and 47 2% of the time in CH sessions. Receivers actions also correlate with these classes of signal combinations. In Prisoners Dilemma and Stag Hunt, receivers respond to truth by choosing an action the same as the sender s current-round message; this happens with frequency 56 2% in PD sessions and 94 6% in SH sessions, though only 51 9% in CH sessions. Receivers in Prisoners Dilemma and Stag Hunt are substantially more likely to choose C following a nice lie than following a damned lie. In PD sessions, they choose C with frequency 52 4% following a nice lie but only 17 3% following a damned lie ; in SH sessions, the frequencies are 80 0% and 46 2%. In Chicken, this pattern does not hold; receivers choose C with frequency 68 3% following a nice lie, and 72 2% following a damned lie. These numbers, while suggestive, should be viewed with caution due to small sample sizes in many cases. In order to draw solid conclusions, we estimate another set of probits using data from the WDL cells only. As in Table 6, the dependent variable is whether a player chose C, and we again estimate coefficients separately for senders and receivers and for each of the three games. In addition to the two-signal dummy variables that we considered in our earlier regressions, we add four three-signal dummies, CCC, CCD, CDC, and CDD, equal to 1 if the (previousround message, current-round message, previous-round action) combination is CCC, CCD, CDC, or CDD. Notice that each of these combinations corresponds to the addition of a C previous-round message to one of the two-signal combinations; including these particular signal combinations allows us to evaluate the incremental contribution of the extra piece of information provided in 20. Additionally, we could break down truth into happy truth (C previous-round message and action) and bitter truth (D previous-round message and action). Overall, senders are likely to be truthful in the current round following either kind of truthfulness in the previous round; however, there are some differences between play following happy truth and play following bitter truth. Most notably, senders in the PD sessions are truthful 83 8% of the time following happy truth but only 64 8% of the time following bitter truth (moreover, C messages following bitter truth are truthful only 26 4% of the time). Differences are smaller in the other two games; in SH sessions, senders are truthful 98 2% of the time following happy truth and 89 7% of the time following bitter truth, while in CH sessions, senders are truthful with frequency 85 8% following happy truth and 96 8% following bitter truth. Since the differences in subsequent play between happy truth and bitter truth are smaller than those between nice lie and damned lie, we combine the two types of truth in the analysis here and later.